If , then x is

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{6}$

12.

If $\sqrt{\mathrm{r}} = {\mathrm{ae}}^{\mathrm{\theta cot}\left(\mathrm{\alpha }\right)}$ where a and $\mathrm{\alpha }$, are real numbers, then $\frac{{\mathrm{d}}^2\mathrm{r}}{{\mathrm{d\theta }}^2} - 4{\mathrm{rcot}}^2\left(\mathrm{\alpha }\right)$ is

• r

• $\frac{1}{\mathrm{r}}$

• r

• 0

The derivative of ${\tan }^{-1}\left[\frac{\sin \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}\right]$ with respect to ${\tan }^{-1}\left[\frac{\cos \left(\mathrm{x}\right)}{1 + \sin \left(\mathrm{x}\right)}\right]$ is

• 2

• - 1

• 0

• - 2

$\frac{\mathrm{d}}{\mathrm{dx}}\left[\cos \left({\cot }^{-1}\left(\sqrt{\frac{2 +\hspace{0.17em}\mathrm{x}}{2 - \mathrm{x}}}\right)\right)\right]$ is

• $\frac{1}{4}$

• $\frac{1}{2}$

• $- \frac{1}{2}$

• $- \frac{3}{4}$

$\mathrm{If} {\cos }^{-1}\left(\frac{\mathrm{y}}{\mathrm{b}}\right) = \mathrm{nlog}\left(\frac{\mathrm{x}}{\mathrm{n}}\right)$, then

• ${\mathrm{xy}}_1 = \mathrm{n}\sqrt{{\mathrm{b}}^2 - {\mathrm{y}}^2}$

• ${\mathrm{xy}}_1 + \mathrm{n}\sqrt{{\mathrm{b}}^2 - {\mathrm{y}}^2} = 0$

• ${\mathrm{y}}_1 = \mathrm{x}\sqrt{{\mathrm{b}}^2 - {\mathrm{y}}^2}$

• ${\mathrm{xy}}_1 - \sqrt{{\mathrm{b}}^2 - {\mathrm{y}}^2} = 0$

A sphere increases its volume at the rate of $\mathrm{\pi }$ cc/s. The rate at which its surface area increases when the radius is 1 cm is

• $2\mathrm{\pi } \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• $\mathrm{\pi } \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• $\frac{3\mathrm{\pi }}{2} \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• $\frac{\mathrm{\pi }}{2} \mathrm{sq} \mathrm{cm}/\mathrm{s}$

$\left|\begin{array}{ccc}\sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)& \sin \left(\mathrm{\alpha } + \mathrm{\delta }\right)\\ \sin \left(\mathrm{\beta }\right)& \cos \left(\mathrm{\beta }\right)& \sin \left(\mathrm{\beta } + \mathrm{\delta }\right)\\ \sin \left(\mathrm{\gamma }\right)& \cos \left(\mathrm{\gamma }\right)& \sin \left(\mathrm{\gamma } + \mathrm{\delta }\right)\end{array}\right|$ is equal to

• 0

• 1

• $1 + \sin \left(\mathrm{\alpha }\right)\sin \left(\mathrm{\beta }\right)\sin \left(\mathrm{\gamma }\right)$

• $1 - \left(\sin \left(\mathrm{\alpha }\right) - \sin \left(\mathrm{\beta }\right)\right)\left(\sin \left(\mathrm{\beta }\right) - \sin \left(\mathrm{\gamma }\right)\right)\left(\sin \left(\mathrm{\gamma }\right) - \sin \left(\mathrm{\alpha }\right)\right)$

If a, b and c are unit vectors such that a + b + c = 0, then angle between a and b is

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{2\mathrm{\pi }}{3}$

• $\mathrm{\pi }$

If a, b and c are non-coplanar, then the value of $\mathrm{a} . \left\{\frac{\mathrm{b} \times \mathrm{c}}{3\mathrm{b} . \left(\mathrm{c} \times \mathrm{a}\right)}\right\} - \mathrm{b} . \left\{\frac{\mathrm{c} \times \mathrm{a}}{2\mathrm{c} . \left(\mathrm{a} \times \mathrm{b}\right)}\right\}$ is

• $- \frac{1}{2}$

• $- \frac{1}{3}$

• $- \frac{1}{6}$

• $\frac{1}{6}$

If 2i + 3j, i + j + k and $\mathrm{\lambda }$i + 4j + 2k taken in an order are coterminous edges of a parallelopiped of volume 2 cu units, then value of $\mathrm{\lambda }$ is

• - 4

• 2

• 3

• 4